Axiom ax-mulass 8272

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 8230. Proofs should normally use mulass 8300 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 8167	. . . 4 class ℂ
3	1, 2	wcel 2209	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2209	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2209	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1009	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 8174	. . . . 5 class ·
10	1, 4, 9	co 6075	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 6075	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 6075	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 6075	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1402	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  8300